Ext-Multiplicity Theorem for Standard Representations of $({\mathrm{GL}}_{n+1},{\mathrm{GL}}_n)$

Abstract

Let ${\pi }_1$ be a standard representation of ${\mathrm{GL}}_{n+1}(F)$ and let ${\pi }_2$ be the smooth dual of a standard representation of ${\mathrm{GL}}_n(F)$. When $F$ is non-Archimedean, we prove that ${\mathrm{Ext}}_{{\mathrm{GL}}_n(F)}^i({\pi }_1,{\pi }_2)$ is $\cong \mathbb{C}$ when $i=0$ and vanishes when $i\ge 1$. The main tool of the proof is a notion of left and right Bernstein-Zelevinsky filtrations. An immediate consequence of the result is to give a new proof on the multiplicity at most one theorem. Along the way, we also study an application of an Euler-Poincaré pairing formula of D. Prasad on the coefficients of Kazhdan-Lusztig polynomials. When $F$ is an Archimedean field, we use the left-right Bruhat-filtration to prove a multiplicity result for the equal rank Fourier-Jacobi models of standard principal series.

arXiv comment23 pages, v2: 25 pages, minor changes, v3: close to published version

References in corpus (6)

• Schwartz functions on Nash manifolds
• Branching laws for Classical Groups: the non-tempered case
• La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires
• The Real Chevalley Involution
• Gross--Prasad periods for reducible representations
• Multiplicity one theorem for $({\mathrm{GL}}_{n+1},{\mathrm{GL}}_n)$ over a local field of positive characteristic